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Operator positivity and analytic models of commuting tuples of operators. (English) Zbl 1385.47007
Authors’ abstract: We study analytic models of operators of class $$C_{\cdot 0}$$ with natural positivity assumptions. In particular, we prove that for an $$m$$-hypercontraction $$T \in C_{\cdot 0}$$ on a Hilbert space $$\mathcal H$$, there exist Hilbert spaces $$\mathcal E$$ and $$\mathcal E_*$$ and a partially isometric multiplier $$\theta \in \mathcal M(H^2(\mathcal E), A^2_m(\mathcal E_*))$$ such that $\mathcal H \mathcal Q_{{\theta}} = A^2_m(\mathcal E_*) \theta H^2(\mathcal E) \text{ and } T P_{\mathcal Q_\theta} M_z|_{\mathcal Q_\theta},$ where $$A^2_m(\mathcal E_*)$$ is the $$\mathcal E_*$$-valued weighted Bergman space and $$H^2(\mathcal E)$$ is the $$\mathcal E$$-valued Hardy space over the unit disc $$\mathbb{D}$$. We then proceed to study analytic models for doubly commuting $$n$$-tuples of operators and investigate their applications to joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over the polydisc. In particular, we completely analyze doubly commuting quotient modules of a large class of reproducing kernel Hilbert modules, in the sense of J. Arazy and M. Engliš [Trans. Am. Math. Soc. 355, No. 2, 837–864 (2003; Zbl 1060.47013)], over the unit polydisc $$\mathbb{D}^n$$.

##### MSC:
 47A45 Canonical models for contractions and nonselfadjoint linear operators 47A13 Several-variable operator theory (spectral, Fredholm, etc.) 47A15 Invariant subspaces of linear operators 47A20 Dilations, extensions, compressions of linear operators 47A80 Tensor products of linear operators 47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 47B38 Linear operators on function spaces (general)
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##### References:
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